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### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 15: Power series function representation

# Finding function from power series by integrating

When a power series S₁ is an antiderivative of a geometric series S₂, we can find the function represented by S₁ by integrating the expression for S₂.

## Want to join the conversation?

• Isn't the answer ln |1-2x| -2?
• No, g'(0)= - 2, but g(0)=0, when we substitute 0 to the second given function. This way c=0 and g(x)=ln |1-2x|
• I would really love to see some context for this massive unit....It's all good to understand how to calculate the interval of convergence for geometric series and power series and their derivatives and integral functions......but what is it good for?
(1 vote)
• Why do we substitute x with 0 to find C?
(1 vote)
• We need to find C in order to determine what the function actually is. We know that the equation holds for all x, so we can plug in any x we like to get an equation where we can solve for C. We choose x=0 because it makes almost everything vanish, so solving for C is easier.
• isn't the other function the infinite series -2x/n*(1-2x)?
• You can only do that geometric series trick when the ratio is constant. it's called the common ratio for a reason. Here the ratio depends on n, the term number, which isn't constant (r=2x*(n-1)/n). You can however express it as: Sigma(n=1->infinity) (-(2x)^n/n), which is the Maclaurian series for ln|1-2x| which is our answer
(1 vote)
• I've had this problem in partial fraction expansion before. Suppose all capital letters are constants, and the expression:
x + A = Cx^3 + 4

Is true for all x. Then apparently we can sub a convenient x value, like x = 0, and get that A = 4? I don't get the logic of this; A = 4 makes both sides equal when x = 0, but why does it work for all other x as well? I would appreciate any help!
• You're right, it isn't logical. That's because the supposition itself cannot be true. No values for A and C exist that are constant, independent of x, and satisfy the conditions for ALL x. so this is unsolvable. The closest thing to a solution is A=4, C=1/x^2, but C here isn't a constant obviously
(1 vote)
• for example, if i substitute x=3 both sides, i didn't find same values for the answer